Digital halftoning is a technique for displaying a picture on a two-dimensional medium, in which small dots and a limited number of colors are used. The picture appears to consist of many colors when viewed from a proper distance. For example, a picture consisting of black and white dots can appear to display various grey levels.
Digital printers, which were initially purely black and white machines with a very coarse resolution, have evolved to accommodate colors, finer resolutions, and more recently, more than one bit of information per pixel. A pixel, as described herein, may refer to a pixel or a pel.
The fastest and most commonly used methods for digital halftoning involve dithering algorithms, which use threshold arrays, and are also referred to as dither matrices or dither masks. A dither mask, defined either in hardware or in software, can be thought of as a matrix of numbers. The original forms of these masks used periodic patterns of threshold values, referred to as an “ordered dither.” In a “random dither,” threshold values are taken according to some stochastic process. Conventional dither masks did not provide many grey levels. This was especially true in the case of clustered masks, where darker greys were produced by printing larger clusters of black dots, as opposed to the spreading techniques used in accordance with dispersed masks. Both types of masks are important in digital printing, with clustered masks usually utilized with laser printers, and dispersed masks usually utilized with ink jet printers.
A mask is defined as periodic if the size of the mask is approximately the minimum size required for a given number of distinct grey levels to be rendered, or if the arrangement of threshold values within the mask purposely forms periodic structures. Aperiodic usually refers to a “long period,” which means that each of the threshold values is repeated a relatively large number of times in the mask in a manner in which the resulting mask is not simply a juxtapositioning of smaller identical masks.
Too much randomness in the design of a dither mask blurs the image and yields unaesthetic results. In the late 1980's, it was discovered that blue noise, or noise with the low frequencies attenuated in the radially averaged power spectrum for the vast majority of grey levels, provides improved visual effects, see, e.g., R. Ulichney, “Dithering with Blue Noise,” Proc. IEEE 76, No. 1, 1988, pp. 56-79. Methods to construct dither masks with blue noise were subsequently proposed, see, e.g., U.S. Pat. No. 5,111,310 to Parker et al.; M. Yao et al., “Modified Approach to the Construction of a Blue Noise Mask,” J. of Electronic Imaging 3, No. 1, 1994, pp. 92-97; and R. Ulichney, “The Void-and-Cluster Method for Dither Array Generation,” Proc. SPIE 1913, 1993, pp. 332-343.
Blue noise masks generate dispersed dots, which means that black dots can only cluster if the grey level is dark enough to make it improbable or impossible to have all black dots isolated. As a consequence, blue noise masks are not practical for laser printers or xerographic printers.
To achieve clustering, a traditional threshold array may be constructed so that increasing the grey level corresponds to printing larger clusters at a fixed periodicity. While this method does not produce unpleasant artifacts, either the number of grey levels that may be represented using such a method is too small, or the clusters which are generated are too big. To correct these effects, a multicell array may be used, which is tiled by several single-cluster threshold arrays. In a multicell array, several clusters are grown with the same spatial period as in the single-cluster array, but they are not grown simultaneously. This allows for additional intermediate grey levels. Traditional multicell arrays are considered periodic and the order and manner in which the cells are grown commonly generates unpleasant periodic patterns at several grey levels.
Two important problems in dither mask generation are the production of clustered aperiodic masks and multicell masks with no artifacts. Typically, a multicell clustered dither array, when tested on all possible levels of uniform greys, will generate both good and less acceptable patterns, depending on the uniform grey level to be rendered and the physical characteristics of the printer.
These problems were addressed successively in:                1) U.S. Pat. No. 5,917,951 to Thompson et al. (hereafter referred to as “TTW1”);        2) U.S. Pat. No. 6,025,930 to Thompson et al. (hereafter referred to as “TTW2”);        3) U.S. Pat. No. 6,597,813 to Stanich et al. (hereafter referred to as “STTW”); and        4) C. W. Wu et al., “Supercell dither masks with constrained blue noise interpolation,” NIP 17: IS&T's International Conference on Digital printing Technologies, 2001, pp. 487-490.        
Each patent and paper is incorporated herein by reference. In solving these problems, it was common to provide techniques which combine the advantages of blue noise and clustering, in order to produce an acceptable pattern of halftoning as needed, for instance, in laser or xerographic printers.
More specifically, according to TTW1, TTW2, and STTW, methods of halftoning of grey scale images utilize a pixel-by-pixel comparison of the image against a clustered aperiodic mask comprised of a partly random and partly deterministic single valued function, which is designed to produce visually pleasing dot configurations when thresholded at any level of grey. The basic improvement of TTW2 over TTW1 was to provide more blueish effects. The basic improvement of STTW over TTW1 and TTW2 was the modulation of the clustering when constructing the mask from lightest to darkest grey levels. Thus, nonrandom dither masks are constructed, which present a similar lack of periodicity which makes blue noise pleasant to the eye. However, TTW1, TTW2, and STTW, which provide flexibility and adaptability to aesthetic criteria in the clustered case, fail to provide similar advantages in the dispersed case. More specifically, they fail to provide flexibility in maintaining desired characteristics of undesirable patterns in the dither mask.